The structure, function, and interactions of proteins produce evolutionary patterns that are imprinted on protein sequences. Here, mitochondrial and nuclear sequences will be used to study evolutionary processes and develop understanding of how proteins evolve in the context of structural, energetic, and functional constraints. Improved models of protein evolution will be developed and informed by this deeper understanding, and their utility in predicting mutational effects and structural features will be evaluated. They will also be used to better predict adaptiv bursts and levels of convergence and coevolution among residues, particularly in multigene families. This research is motivated by insights from previous research. First, it is expected from evolutionary simulations that substitution probabilities at individual positions in a protein fluctuate in time due to epistasis (interactions with substitutions at other sites in the same or other proteins). These expectations are supported by strong evidence that substitution processes do regularly fluctuate with time in real proteins. However, current models of protein evolution do not usually allow substitution processes to fluctuate with time, and levels of amino acid convergence in proteins deviate substantially from expectations for such models. Because of this, incorporating such fluctuations is a key feature of the proposed models. Second, current approaches that incorporate structure into evolutionary studies tend to use de novo prediction or pseudo energy potentials to predict the acceptability of substitutions, but these methods are not especially accurate for evolutionary analysis, which includes sequences that have diverged substantially from the sequences of known protein structures. To account for this, rather than allowing such predictions to stand alone, they will be incorporated probabilistically into empirica substitution models to varying degrees depending on expected predictive accuracy and distance from any sequences with known structure. Third, a Bayesian approach to building complex evolutionary models was recently developed that is designed to allow relatively easy computation of processes that fluctuate among sites and over time. This approach using what is called partial sampling of substitution histories makes the proposed methodology feasible. It is expected that the proposed study will make significant improvements in understanding of molecular evolution and how it relates to structure and function. One expected result of this study will be better predictions of mutational effects, which will lead to an improved ability to identify disease-causing mutations in human genome and exome sequencing studies. It is further expected that predictions of structural features when they are unknown will be improved, and researchers will be able to better understand how ancestral functional changes in proteins have arisen through adaptive sequence change.